Abstract
Permutations can represent search problems when all points in the search space have unique evaluations. Given a particular set of N evaluations we have N! search algorithms and N! possible functions. A general No Free Lunch result holds for this finite set of N! functions. Furthermore, it is proven that the average description length over the set of N! functions must be O(N lg N). Thus if the size of the search space is exponentially large with respect to a parameter set which specifies a point in the search space, then the description length of the set of N! functions must also be exponential on average. Summary statistics are identical for all instances of the set of N! functions, including mean, variance, skew and other r-moment statistics. These summary statistics can be used to show that any set of N! functions must obey a set of identical constraints which holds over the set of Walsh coefficients. This also imposes mild constraints on schema information for the set of N! functions. When N = 2 L subsets of the N! functions are related via Gray codes which partition N! into equivalence classes of size 2L.
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References
L. Barbulescu, J.P. Watson, and D. Whitley. Dynamic representations and escaping local optima. In 17th National Conf. on Artificial Intelligence (AAAI), 2000.
J. Culberson. On the Futility of Blind Search. Evolutionary Computation, 6(2):109–127, 1999.
D. Goldberg. Genetic Algorithms and Walsh Functions: Part I, A Gentle Introduction. Complex Systems, 3:129–152, 1989.
D. Goldberg. Genetic Algorithms and Walsh Functions: Part II, Deception and its Analysis. Complex Systems, 3:153–171, 1989.
D. Goldberg and M. Rudnick. Genetic algorithms and the variance of fitness. Complex Systems, 5(3):265–278, 1991.
R. Heckendorn, S. Rana, and D. Whitley. Polynomial time summary statistics for a generalization of MAXSAT. GECCO-99, Morgan Kaufmann, 1999.
R. Heckendorn, S. Rana, and D. Whitley. Test Function Generators as Embedded Landscapes. In Foundations of Genetic Algorithms FOGA-5. Morgan Kaufmann, 1999.
S.A. Kauffman. Adaptation on Rugged Fitness Landscapes. Lectures in the Science of Complexity, pages 527–618. Addison-Wesley, 1989.
N.J. Radcliffe and P.D. Surry. Fundamental limitations on search algorithms: Evolutionary computing in perspective. Lecture Notes in Computer Science 1000. Springer-Verlag, 1995.
S. Rana and D. Whitley. Bit Representations with a Twist. 7th International Conf. on GAs (ICGA), Morgan Kaufmann, 1997.
S. Rana and D. Whitley. Search, representation and counting optima. IMA Workshop on Evolutionary Algorithms. Springer-Verlag, 1998.
C. Reeves and C. Wright. An Experimental Design Perspective on Genetic Algorithms. In FOGA-3, pages 7–22. Morgan Kaufmann, 1995.
C.A. Tovey. Hill climbing and multiple local optima. SIAM Journal on Algebraic and Discrete Methods, 6(3):384–393, July 1985.
D.H. Wolpert and W.G. Macready. No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation, 4:67–82, 1997.
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Whitley, D. (2000). Functions as Permutations: Regarding No Free Lunch, Walsh Analysis and Summary Statistics. In: Schoenauer, M., et al. Parallel Problem Solving from Nature PPSN VI. PPSN 2000. Lecture Notes in Computer Science, vol 1917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45356-3_17
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DOI: https://doi.org/10.1007/3-540-45356-3_17
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